Asymptotic behavior of quasi-nonexpansive mappings. (Q2768010)

This article deals with iterations of quasi-nonexpansive mappings of a nonempty closed convex subset \(K\) in a Banach space \(X\). Let \(\mathcal M\) be the set of all mappings \(T:K\to K\) which are bounded on bounded subsets of \(K\) with the uniformity generated by the base NEWLINE\[NEWLINE \bigl\{(T_1,T_2): \| T_1x-T_2x \|\leq \varepsilon\text{ for all }x \in K\text{ satisfying }\| x\|\leq N \bigr\},\quad N,\varepsilon>0.NEWLINE\]NEWLINE For each nonempty closed convex subset of \(K\), set \({\mathcal M}^F= \{T\in {\mathcal M}: Tx=x\) for all \(x\in F\) and \(D_f(z,Tx)\leq D_f (z,x)\) for all \(z\in F\) and all \(x\in K\}\), where \(f:X\to\mathbb R\cup \{\infty\}\) is a convex function with nonempty algebraic interior and \(D_f(y,x)= f(y)-f(x)-\partial f(x,x-y)\). The main result of the present paper is the following: under certain assumption on \(f\), the iterates of a generic mapping in these spaces converge strongly to a retraction on \(F\).NEWLINENEWLINEFor the entire collection see [Zbl 0971.00058].